Long-range behavior of a nonlocal correlation-energy density functional based on the random-phase approximation

We propose a density-based nonlocal correlation energy functional, corresponding to a revised form of the seamless van der Waals (vdW) density functional developed by Dobson and coauthors in the late nineties. The functional, termed as rDW99 in this work, has the same energy expression as the random phase approximation (RPA), but its key ingredient---the noninteracting density response function---is expressed entirely in terms of the electron density and density gradient. This is enabled by generalizing the Lindhard function from the homogeneous electron gas (HEG) to inhomogeneous electron systems. Compared to the original DW99 functional of Dobson and coauthors, we have introduced two revisions in rDW99: (1) A gap correction is incorporated into the generalized Lindhard function, making the functional more suitable for describing inhomogeneous, and in particular insulating, systems; (2) an approximate, yet fairly accurate, analytical form of the Lindhard function in real space is developed which facilitates the numerical evaluation of the rDW99 functional. In contrast with previously developed vdW density functionals, the nonlocal polarization effect is naturally captured in the rDW99 functional. The functional form, as it stands right now, is computationally still rather involving in a straightforward implementation. However, we anticipate that numerical techniques can be developed to significantly reduce the computational cost. As a first step, we assess the quality of rDW99 for describing the vdW interactions in the nonoverlapping asymptotic regime by computing the ${C}_{6}$ coefficients for a set of 43 atoms and molecules. With only one adjustable parameter, the mean absolute percentage error (MAPE) of the ${C}_{6}$ coefficients is estimated to be $8.7%$. In this work, ${C}_{6}$ coefficients corresponding to the standard RPA method are also computed for the same test set, and the MAPE for the RPA ${C}_{6}$ coefficients is estimated to be $13.2%$.